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Properties

Label 2-531-3.2-c4-0-0
Degree $2$
Conductor $531$
Sign $-0.816 - 0.577i$
Analytic cond. $54.8894$
Root an. cond. $7.40874$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.71i·2-s − 6.25·4-s + 39.7i·5-s − 15.3·7-s − 45.9i·8-s + 187.·10-s + 168. i·11-s − 143.·13-s + 72.4i·14-s − 316.·16-s − 26.0i·17-s + 401.·19-s − 248. i·20-s + 793.·22-s + 224. i·23-s + ⋯
L(s)  = 1  − 1.17i·2-s − 0.390·4-s + 1.58i·5-s − 0.313·7-s − 0.718i·8-s + 1.87·10-s + 1.38i·11-s − 0.850·13-s + 0.369i·14-s − 1.23·16-s − 0.0900i·17-s + 1.11·19-s − 0.620i·20-s + 1.63·22-s + 0.424i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $-0.816 - 0.577i$
Analytic conductor: \(54.8894\)
Root analytic conductor: \(7.40874\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{531} (296, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 531,\ (\ :2),\ -0.816 - 0.577i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.06430253711\)
\(L(\frac12)\) \(\approx\) \(0.06430253711\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 + 453. iT \)
good2 \( 1 + 4.71iT - 16T^{2} \)
5 \( 1 - 39.7iT - 625T^{2} \)
7 \( 1 + 15.3T + 2.40e3T^{2} \)
11 \( 1 - 168. iT - 1.46e4T^{2} \)
13 \( 1 + 143.T + 2.85e4T^{2} \)
17 \( 1 + 26.0iT - 8.35e4T^{2} \)
19 \( 1 - 401.T + 1.30e5T^{2} \)
23 \( 1 - 224. iT - 2.79e5T^{2} \)
29 \( 1 + 1.43e3iT - 7.07e5T^{2} \)
31 \( 1 + 552.T + 9.23e5T^{2} \)
37 \( 1 - 927.T + 1.87e6T^{2} \)
41 \( 1 - 856. iT - 2.82e6T^{2} \)
43 \( 1 + 1.23e3T + 3.41e6T^{2} \)
47 \( 1 + 2.24e3iT - 4.87e6T^{2} \)
53 \( 1 + 1.63e3iT - 7.89e6T^{2} \)
61 \( 1 + 5.43e3T + 1.38e7T^{2} \)
67 \( 1 + 6.11e3T + 2.01e7T^{2} \)
71 \( 1 + 1.42e3iT - 2.54e7T^{2} \)
73 \( 1 + 7.01e3T + 2.83e7T^{2} \)
79 \( 1 + 433.T + 3.89e7T^{2} \)
83 \( 1 - 2.16e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.39e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.29e4T + 8.85e7T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.52984287681426081075021674385, −9.865520975407818023206271669143, −9.546582194732691624047880981510, −7.57016868493309253128353942959, −7.13328665014338857258604683017, −6.16566666943475032918118848081, −4.60652507336540442303360571712, −3.44081263640453773258777403493, −2.66916297721987760464018860491, −1.81547352673483139194460523579, 0.01583219890856980759615204323, 1.31743881843144440261175043516, 3.06666871034777508804550480830, 4.62394968461491892020847356957, 5.36499047312853871845415357781, 6.05524301439941483483933989429, 7.24140974681571264404688739998, 8.059785965884201783144031667119, 8.841555168878961363332526788547, 9.383587998014044139739538451443

Graph of the $Z$-function along the critical line